Correction of Line-Sampling Bias of Rock Discontinuity Orientations Using a Modified Terzaghi Method

)e Terzaghi method is widely used to correct the line-sampling bias of rock discontinuity orientations.)emethod includes four procedures, one of which is meshing the stereographic projection diagram into cells. )e method is based on the biascompensatory factor, 1/sin θ, where θ is the angle between the scanline and the discontinuity defined at each cell center.)is paper presents a modified Terzaghi method that eliminates meshing, thereby reducing the method to three steps that (1) count the frequencies, (2) weigh the frequencies by the bias-compensatory factor, and (3) round the weighed frequencies to the nearest integer. Due to the elimination of the mesh, the counting object has changed to the frequency at each pole, and θ in the biascompensatory factor is redefined as the angle between the scanline and the discontinuity at each pole. )e applicability of the redefined bias-compensatory factor is verified through amathematical logical deduction.)e accuracy of the conventional and the modified Terzaghi methods are compared using a case study in Wenchuan, China, revealing improved accuracy for the latter.


Introduction
Many important engineering structures are built on or in rock, such as tunnels, dam foundations, bridge foundations, tall buildings, etc. Rock is an inhomogeneous natural material owing to the presence of discontinuities, defined as any significant mechanical break or fracture of negligible tensile strength [1]. Discontinuities include bedding planes, faults, fissures, fractures, joints, etc. e orientation of discontinuities is known to greatly affect the kinematical and mechanical behavior of rock [2][3][4][5]. ese orientations can be observed on rock exposures such as outcrops, tunnel faces, pit faces, and drillhole faces. A common observational method is line sampling down a borehole or along a scanline on an exposure. However, such sampling biases the apparent abundance of the orientations, because the probability that a particular discontinuity is intersected by a scanline depends on its orientation [6,7]. To correct for this bias, Terzaghi [8] proposed a well-known method that weighs the observed frequencies by a bias-compensatory factor, which is the reciprocal of the sine of the intersection angle between the scanline and discontinuity. A detailed description of this method is available in Einstein and Beacher [9,10]. e Terzaghi method has been widely used by researchers, including Goodman [11], Park [12], and Fouché and Diebolt [13].
Tang et al. [14] found that the Terzaghi method involves an error, even if optimizing countermeasures are applied. Unfortunately, the origin of the error is unknown, but the meshing procedure is suspected to be a significant component. is paper presents a modified Terzaghi method that eliminates the meshing procedure. Following a brief introduction of the conventional Terzaghi method, we describe this modified method and its internal procedures.
Owing to mesh elimination, some procedural modifications are required. Second, the bias-compensatory factor had been reobtained in the case of the presence of mesh [8], so we determine whether this factor is still applicable in the nomesh case, using a mathematical logical deduction. Finally, we apply both the conventional and the modified Terzaghi methods to a test case and compare the accuracy of the two results.

Procedures.
e procedures of the conventional Terzaghi method [8] are (Figure 1(a)) as follows: (1) Subdivide the projection net into cells.
(2) Count the frequencies lying in each cell.
(3) Weigh the frequencies by the bias-compensatory factor 1/sin θ, where θ is the intersection angle between the scanline and the discontinuity defined at each cell center. Since the orientation is composed of two elements, that is, the dip direction and dip angle, the weighing should be executed for dip direction and dip angle, respectively. (4) Because the frequencies defined in this manner must be integers, the weighed frequencies that are rounded to the nearest integer.
e procedures of the modified Terzaghi method are (Figure 1(b)) as follows: (1) Count the frequencies at each pole. Owing to mesh elimination, the counting objects have changed to the frequencies at each pole. (2) Weigh the frequencies by the bias-compensatory factor 1/sin θ, where θ is redefined as the angle between the scanline and the discontinuity at each pole. (3) Round the weighed frequencies to the nearest integer.

Applicability of the Redefined Bias-Compensatory
Factor. e kernel of the modified Terzaghi method is the biascompensatory factor. is factor is known to be 1/sin θ for the conventional Terzaghi method [8], but its applicability to the modified Terzaghi method with redefinition remains unknown. In this section, a mathematical deduction is used to verify its applicability.
For the conventional Terzaghi method, the weighing procedure is obtained by multiplying the bias-compensatory factor, 1/sin θ, by the observed frequency defined in the cell [8]. is step can be formulated as follows: where α is the dip direction, β is the dip angle, ∆α is the cell size on dip direction, ∆β is the cell size on dip angle ( Figure 2), P A is the corrected frequency in cell, p AB (α, β) is the joint probability density that the dip direction and dip angle are intersected by scanline, and P is the observed frequency in cell. For the modified Terzaghi method, the weighing procedure is to multiply 1/sin θ by the observed frequency at each orientation pole. Such a pole, in calculus, can be regarded as a special cell, that is, an infinitesimal cell around the pole (∆α ⟶ 0°, ∆β ⟶ 0°). In this case, P and P A in Equation (1) become the observed frequency and the corrected frequency at the orientation pole, respectively. erefore, despite a slight redefinition of θ, the biascompensatory factor can be also used for the modified Terzaghi method.

Application of the Modified Terzaghi Method
e use of the modified Terzaghi method is illustrated and its accuracy is tested against the conventional Terzaghi method, using a real example of bedding orientation observations in a roadcut in China. e study area is near Yingxiu town in Wenchuan, Sichuan Province, and located only about 1,800 m east of the epicenter of the 2008 Wenchuan Earthquake ( Figure 3). e particular roadcut is 11 m long, 5 m wide, and 6 m high and consists of Upper Triassic lithic arkose of the Xujiahe Formation. e rock has two discontinuity sets, one of which is the bedding plane.
A scanline with the trend/plunge of 108/15°was fixed to observe the bedding planes on this outcrop ( Figure 4). Table 1 lists 55 observed orientations, whose pole diagram is shown in Figure 5.
First, the sampling bias of the observed orientations was corrected according to the conventional and the modified Terzaghi methods. A cell size of 2°× 2°was selected for the conventional Terzaghi method because Tang [15] concluded that this size optimizes the accuracy. e result corrected by the conventional Terzaghi method is shown in Figure 6(a) and the result corrected by the modified Terzaghi method is shown in Figure 6(b). In addition, the volumetric abundance, diameter, and aperture were calculated, with the results listed in Table 2.
Next, a three-dimensional model of the rock was constructed by discrete fracture network modeling, as described by Xu and Dowd [16], Brzovic and Villaescusa [17], and Grenon and Hadjigeorgiou [18]. Multiplying the volumetric abundance (10 m −3 ) by the volume of the simulated zone (400 m 3 ), in Table 2, results in a total of 4,000 discontinuities. en, pseudorandom numbers of these 4,000 discontinuities were generated for the five elements, namely the X-coordinate, Y-coordinate, Z-coordinate, diameter and aperture.
ese pseudorandom numbers are not listed here because of space limitations. After entering the pseudorandom numbers and the corrected orientation data into modeling software such as OpenGL or AutoCAD, two models can be built corresponding to the conventional and the modified Terzaghi methods (Figure 7). A scanline with the same orientation as the field scanline was applied to the model outcrop and the discontinuities that are intersected by  Advances in Civil Engineering this scanline were then "measured." e quantity of these measured discontinuities is set equal to the number of observed discontinuities in the field. To distinguish between these measured discontinuities and the real discontinuities observed in the field, the former are named "modeled" discontinuities. Figure 8 shows the modeled discontinuity orientations. ird, the distribution difference between the observed and the modeled orientations was tested by the Kolmogorov-Smirnov two-sample test. is nonparametric hypothesis test evaluates the difference between the cumulative distribution functions of two sample data vectors. e test can be executed by the software Statistical Product and Service Solutions and returns an asymptotic significance to characterize the difference. e significance ranges from 0 to 1; the higher the significance, the lower the difference. More information about this test is given inÖzcomak et al. [19] and Al-Labadi and Zarepour [20]. e test returned two significances corresponding to the dip direction and dip angle. ese two significances were combined and their average is 0.686 for the conventional Terzaghi method and 0.988 for the modified Terzaghi method. It is apparent that the latter value is much higher than the former value, demonstrating that the modified Terzaghi method is more accurate than the conventional method, even if the optimal cell size of 2°× 2°is chosen for the conventional method.

Discussion
Tang [14] presented the accuracy of the conventional Terzaghi method in two cases: Case 2. If n does not approach ∞,

Advances in Civil Engineering
where D is a given orientation interval and p A (α, β) is the joint probability density of the dip direction and dip angle in the rockmass.
As mentioned in Section 2.2, the elimination of mesh in the modified Terzaghi method is identical to meshing the projection diagram into infinitesimal cells (Case 1 for Equation (2)). Tang [14] proved that Equation (2) is accurate, while Equation (3) introduces error. Hence, the modified Terzaghi method is a more accurate procedure. In contrast, for the conventional Terzaghi method, the cells can only approach but can never attain an infinitesimal size. Such a mesh can only meet Case 2 and not Case 1, so the conventional Terzaghi method will consequently involve error. is analysis reinforces the result regarding accuracy obtained in Section 3.
As shown in Section 3, the significance of the modified Terzaghi method approaches but does not attain unity, suggesting that factors in addition to meshing contribute error to the Terzaghi method. One inferred source of error is rounding the corrected frequencies to the nearest integer.
is issue will be studied in the future.

Conclusions
e Terzaghi method is widely used to correct the observed distribution of rock discontinuity orientations. A modified Terzaghi method that eliminates the meshing procedure was developed to improve accuracy. is modified method only includes three procedures that (1) count the frequencies at each pole, (2) weigh the observed frequencies by the biascompensatory factor, and (3) round the weighed frequencies to the nearest integer. Mesh elimination requires that the counting objects are changed to the frequencies at each pole instead of the frequencies lying in each cell; accordingly, the bias-compensatory factor has become the reciprocal of the sine of the intersection angle between the scanline and the discontinuity at each pole, rather than the reciprocal of the sine of the intersection angle between the scanline and the discontinuity defined at each cell center. A mathematical logical deduction verifies that this slightly redefined biascompensatory factor can be used in the modified Terzaghi method. e conventional and the modified Terzaghi methods were both applied to a case study near Yingxiu, in Sichuan, China.
e results indicate that the modified Terzaghi method performed more accurately than the conventional method.

Disclosure
Mutasim Adam Mohamed Ez Eldin is presently at the Section of Engineering Geology, Department of Applied Geology, Saudi Geological Survey, Jeddah 21514, Saudi Arabia.